To derive the consistency of \(\hat{\beta }_{\mathrm {AL}}\) , a careful look at the proof in Appendix  shows that the condition \(a_n \sqrt{{\mu _n s_n}/p_n} \rightarrow 0\)  could be sufficient. The convergence rate, i.e. \(O_{\mathrm {P}} ( \sqrt{p_n/\mu _n})\) , is \(\sqrt{p_n}\)  times the convergence rate of the estimator obtained when \(p_n\)  is constant [1]}. It also corresponds to the rate of convergence obtained by [2]} for generalized linear models when \(p_n\rightarrow \infty \)  and when \(\mu _n\)  corresponds to the standard sample size. It is worth pointing out that a possible diverging number of non-zero coefficients \(s_n\)  does not affect the rate of convergence. It does however impose a more restrictive condition on \(a_n\) . We also emphasize the fact that in the unregularized situation, i.e. when \(\lambda _{n,j}=0\)  for \(j=1,\dots ,p_n\) , Theorem REF  (i) remains valid. Hence a simple choice for the initial estimate \(\widetilde{\beta }\)  defining the ALDS estimate is the unregularized estimator, i.e. the maximum of the Poisson likelihood given by (REF ).
